Finite Group DatabasebetaFinite Group DatabasebetaOrder 3072.
At least 6440 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has at least 6440 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#85645 |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C23×C3 |
| Frattini, socle | C23 | C23×C22⋊A4 |
| Fitting | #72240149 | C3 |
| hypercenter | D4 | #85645 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72241294 | C2 | semidirect |
| 2. #72241282 | C2 | semidirect |
| 3. #72240149 | C3 | semidirect |
| 4. #72233786 | C2 | semidirect |
| 5. #72233218 | C2 | semidirect |
| 6. C2×#35346⋊C3 | C2 | semidirect |
| 7. #11710929 | C22 | semidirect |
| 8. C2×C22⋊#997 | C22 | semidirect |
| 9. #35446⋊C3 | C22 | semidirect |
| 10. #35346⋊C3 | C22 | semidirect |
| 11. #35346⋊C3 | C22 | semidirect |
| 12. C22×H(7,𝔽2) | C2×C3 | semidirect |
| 13. #7630606 | C2×C3 | semidirect |
| 14. #7630442 | C2×C3 | semidirect |
| 15. #7629120 | C2×C3 | semidirect |
| 16. C2×#35346 | C2×C3 | semidirect |
| 18. C22×#585 | C22×C3 | semidirect |
| 19. C23×H(5,𝔽2) | C22×C3 | semidirect |
| 20. C25×D4 | A4 | semidirect |
| 21. D4×#120 | A4 | semidirect |
| 22. #36078 | A4 | semidirect |
| 23. #35446 | C22×C3 | semidirect |
| 25. #35346 | C22×C3 | semidirect |
| 26. C27 | C2×A4 | semidirect |
| 27. C22×2−1+4 | C2×A4 | semidirect |
| 30. C24×D4 | C2×A4 | semidirect |
| 31. C24×D4 | C2×A4 | semidirect |
| 32. C24×D4 | C2×A4 | semidirect |
| 33. C25×C4 | C2×A4 | semidirect |
| 34. C22×#120 | C2×A4 | semidirect |
| 35. #2847 | C2×A4 | semidirect |
| 36. #2846 | C2×A4 | semidirect |
| 37. #2615 | C2×A4 | semidirect |
| 38. #2614 | C2×A4 | semidirect |
| 39. #2388 | C2×A4 | semidirect |
| 40. C2×#393 | C2×A4 | semidirect |
| 41. C4×#120 | C2×A4 | semidirect |
| 45. C26 | C22×A4 | semidirect |
| 46. C22×#41 | C22×A4 | semidirect |
| 47. C23×Q8 | C22×A4 | semidirect |
| 51. C24×C4 | C22×A4 | semidirect |
| 53. C2×#120 | C22×A4 | semidirect |
| 54. #395 | C22×A4 | semidirect |
| 56. #393 | C22×A4 | semidirect |
| 57. #379 | C22×A4 | semidirect |
| 62. C22×D4 | #120⋊C3 | semidirect |
| 67. C24 | C2×#120⋊C3 | semidirect |
| 71. C22×C4 | C2×#120⋊C3 | semidirect |
| 75. C23 | #87547 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72241294 | 2 |
| #72241282 | 2 |
| #72240149 | 3 |
| #72233786 | 2 |
| #72233218 | 2 |
| C2×#35346⋊C3 | 2 |
| C2×D4×C22⋊A4 | 4 |
| D4×#120⋊C3 | 4 |
| #36078⋊C3 | 4 |